convergence rate of euler–maruyama scheme for sdes with ...term b satisﬁes the global...

Journal of Theoretical Probability (2019) 32:848–871https://doi.org/10.1007/s10959-018-0854-9

Convergence Rate of Euler–Maruyama Scheme for SDEswith Hölder–Dini Continuous Drifts

Jianhai Bao2 · Xing Huang1 · Chenggui Yuan2

Received: 6 May 2017 / Revised: 30 March 2018 / Published online: 31 August 2018© The Author(s) 2018

AbstractIn this paper, we are concerned with convergence rate of Euler–Maruyama scheme forstochastic differential equationswithHölder–Dini continuous drifts. The key contribu-tions are as follows: (i) bymeans of regularity of non-degenerate Kolmogrov equation,we investigate convergence rate of Euler–Maruyama scheme for a class of stochasticdifferential equations which allow the drifts to be Dini continuous and unbounded; (ii)by the aid of regularization properties of degenerate Kolmogrov equation, we discussconvergence rate of Euler–Maruyama scheme for a range of degenerate stochasticdifferential equations where the drifts are Hölder–Dini continuous of order 2

3 withrespect to the first component and are merely Dini-continuous concerning the secondcomponent.

Keywords Euler–Maruyama scheme · Convergence rate · Hölder–Dini continuity ·Degenerate stochastic differential equation · Kolmogorov equation

Mathematics Subject Classification (2010) 60H35 · 41A25 · 60H10

1 Introduction andMain Results

In their paper [23], Wang and Zhang studied existence and uniqueness for a class ofstochastic differential equations (SDEs) with Hölder–Dini continuous drifts; Wang

Supported in part by NNSFC (11771326, 11831014).

B Chenggui [email protected]

Jianhai [email protected]

Xing [email protected]

1 Center of Applied Mathematics, Tianjin University, Tianjin 300072, China

2 Department of Mathematics, Swansea University, Singleton Park SA2 8PP, UK

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[22] also investigated the strong Feller property, log-Harnack inequality and gradientestimates for SDEs with Dini-continuous drifts. So far there are no numerical schemesavailable for SDEs with Hölder–Dini continuous drifts. So the aim of this paper isto prove the convergence of Euler–Maruyama (EM) scheme and obtain the rate ofconvergence for these equations under reasonable conditions.

It is well-known that convergence rate of EM for SDEs with regular coefficients isone-half, see, e.g., [11]. With regard to convergence rate of EM scheme under varioussettings, we refer to, e.g., [1] for stochastic differential delay equations (SDDEs)with polynomial growth with respect to (w.r.t.) the delay variables, [4] for SDDEsunder local Lipschitz and monotonicity condition, [14] for SDEs with discontinuouscoefficients, and [25] for SDEs under log-Lipschitz condition, whereas for SDEs withnon-globally Lipschitz continuous coefficients; see, e.g., [2,6–8], to name a few.On theother hand, Hairer et al. [5] have established the first result in the literature that Euler’smethod converges to the solution of an SDEwith smooth coefficients in the strong andnumerical weak sense without any arbitrarily small polynomial rate of convergence,and Jentzen et al. [9] have further given a counterexample that no approximationmethod converges to the true solution in the mean square sense with polynomial rate.

The rate of convergence of EM scheme for SDEs with irregular coefficients hasalso gained much attention. For instance, adopting the Yamada–Watanabe approx-imation approach, [3] discussed strong convergence rate in L p-norm sense; usingthe Yamada–Watanabe approximation trick and heat kernel estimate, [16] studiedstrong convergence rate in L1-norm sense for a class of non-degenerate SDEs, wherethe bounded drift term satisfies a weak monotonicity and is of bounded variationw.r.t. a Gaussian measure and the diffusion term is Hölder continuous; applying theZvonkin transformation, [18] discussed strong convergence rate in L p-norm sense forSDEs with additive noises, where the drift coefficient is bounded and Hölder contin-uous.

It is worth pointing out that [16,18] focused on convergence rate of EM for SDEswith Hölder continuous and bounded drifts, which rules out Hölder–Dini continuousand unbounded drifts. On the other hand,most of the existing literature on convergencerate of EM scheme is concerned with non-degenerate SDEs. Yet the correspond-ing issue for degenerate SDEs is scarce, to the best of our knowledge. So, in thiswork, we will not only investigate the convergence of the EM scheme for SDEs withHölder–Dini continuous drifts, but will also study the degenerate setup. For well-posedness of SDEs with singular coefficients, we refer to, e.g., [13,22,23,27] for moredetails.

Throughout the paper, the following notation will be used. Let n,m be positiveintegers, (Rn, 〈·, ·〉 , |·|) the n-dimensional Euclidean space, andRn⊗R

m the family ofall n×mmatrices. Let ‖·‖ and ‖·‖HS stand for the usual operator norm and theHilbert–Schmidt norm, respectively. Fix T > 0 and set ‖ f ‖T ,∞ := supt∈[0,T ],x∈Rm ‖ f (t, x)‖for an operator-valued map f on [0, T ] × R

m . C(Rm;Rn) means the continuousfunctions f : Rm → R

n . Let C2(Rn;Rn ⊗ Rn) be the family of all continuously

twice differentiable functions f : Rn → Rn ⊗ R

n . Denote Mnnon by the collection of

all nonsingular n×n-matrices. LetS0 be the collection of all slowly varying functionsφ : R+ → R+ at zero in Karamata’s sense (i.e., limt→0

φ(λt)φ(t) = 1 for any λ > 0),

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850 Journal of Theoretical Probability (2019) 32:848–871

which are bounded from 0 and ∞ on [ε,∞) for any ε > 0. Let D0 be the family ofDini functions, i.e.,

D0 : ={φ

∣∣∣φ : R+ → R+ is increasing and∫ 1

0

φ(s)

sds < ∞

}.

A function f : Rm → Rn is called Dini continuity if there exists φ ∈ D0 such that

| f (x)− f (y)| ≤ φ(|x− y|) for any x, y ∈ Rm .We remark that every Dini-continuous

function is continuous and every Lipschitz continuous function is Dini continuous;Moreover, if f is Hölder continuous, then f is Dini continuous. Nevertheless, thereare numerous Dini-continuous functions, which are not Hölder continuous at all, see,e.g.,

φ(x) ={

1(log(c+x−1))(1+δ) , x > 0

0, x = 0

for some constants δ > 0 and c ≥ e3+2δ . Set

D := {φ ∈ D0|φ2 is concave} and Dε := {φ ∈ D |φ2(1+ε) is concave}

for some ε ∈ (0, 1) sufficiently small. Clearly, φ constructed above belongs to Dε. Afunction f : Rm → R

n is called Hölder–Dini continuity of order α ∈ [0, 1) if

| f (x) − f (y)| ≤ |x − y|αφ(|x − y|), |x − y| ≤ 1

for some φ ∈ D0; see, for instance,

f (x) ={

1(1+x)α(log(c+x−1))(1+δ) , x > 0

0, x = 0

for some constants c, δ > 0 and α ∈ (0, 1).Before proceeding further, a few words about the notation are in order. Generic

constants will be denoted by c; we use the shorthand notation a � b to mean a ≤ c b.If the constant c depends on a parameter p, we shall also write cp and a �p b.Throughout the paper, for fixed T > 0, CT > 0, dependent on the quantity T , is ageneric constant which may change from line to line.

1.1 Non-degenerate SDEs with Bounded Coefficients

In this subsection, we consider an SDE on (Rn, 〈·, ·〉 , | · |)

dXt = bt (Xt )dt + σt (Xt )dWt , t > 0, X0 = x, (1.1)

where b : R+×Rn → R

n , σ : R+×Rn → R

n⊗Rn , and (Wt )t≥0 is an n-dimensional

Brownianmotion defined on a complete filtered probability space (�,F , (Ft )t≥0,P).

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With regard to (1.1), we suppose that there exists φ ∈ D such that, for any s, t ∈[0, T ] and x, y ∈ R

n,

(A1) σt ∈ C2(Rn;Rn ⊗ Rn), σt (x) ∈ M

nnon, and

‖b‖T ,∞ +2∑

i=0

‖∇ iσ‖T ,∞ + ‖∇σ−1‖T ,∞ + ‖σ−1‖T ,∞ < ∞, (1.2)

where ∇ i means the i th order gradient operator;(A2) (Regularity of b w.r.t. spatial variables)

|bt (x) − bt (y)| ≤ φ(|x − y|);

(A3) (Regularity of b and σ w.r.t. time variables)

|bs(x) − bt (x)| + ‖σs(x) − σt (x)‖HS ≤ φ(|s − t |).

Without loss of generality, we take an integer N > 0 sufficiently large such that thestepsize δ := T /N ∈ (0, 1). The continuous-time EM scheme corresponding to (1.1)is

dYt = btδ (Ytδ )dt + σtδ (Ytδ )dWt , t > 0, Y0 = X0 = x . (1.3)

Herein, tδ := �t/δ δ with �t/δ the integer part of t/δ.The first contribution in this paper is stated as follows.

Theorem 1.1 Let (A1)–(A3) hold. Then

(E

(sup

0≤t≤T|Xt − Yt |2

))1/2

�T φ(CT√

δ)

for some constant CT ≥ 1.

Under (A1) and (A2), (1.1) admits a unique non-explosive strong solution(Xt )t∈[0,T ]; see, e.g., [22, Theorem 1.1]. In Theorem 1.1, by taking φ(x) = xβ forx ≥ 0 andβ ∈ (0, 1], and inspecting closely the argument of Theorem1.1, the concaveproperty of φ2 can be dropped. Moreover, we have

E

(sup

0≤t≤T|Xt − Yt |2

)�T δβ.

So, our present result covers [18, Theorem 2.13], where the drift is Hölder continuous.In particular, for the setting β = 1, it reduces to the classical result on strong con-vergence of EM scheme for SDEs with regular coefficients; see, e.g., [11] for moredetails.

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1.2 Non-degenerate SDEs with Unbounded Coefficients

As we see, in Theorem 1.1, the coefficients are uniformly bounded, and that the driftterm b satisfies the global Dini-continuous condition [see (A2) above], which seemsto be a little bit stringent. Therefore, concerning the coefficients, it is quite naturalto replace uniform boundedness by local boundedness and global Dini continuity bylocal Dini continuity, respectively.

In lieu of (A1)–(A3), as for (1.1) we assume that, for any s, t ∈ [0, T ] and k ≥ 1,

(A1’) σt ∈ C2(Rn;Rn ⊗ Rn), for every x ∈ R

n, σt (x) ∈ Mnnon, and

|bt (x)| +2∑

i=0

‖∇ iσt (x)‖HS + ‖∇σ−1t (x)‖HS

+ ‖σ−1t (x)‖HS ≤ KT (1 + |x |), x ∈ R

n

for some constant KT > 0;(A2’) (Regularity of b w.r.t. spatial variables) There exists φk ∈ D such that

|bt (x) − bt (y)| ≤ φk(|x − y|), |x | ∨ |y| ≤ k;

(A3’) (Regularity of b and σ w.r.t. time variables) For φk ∈ D such that (A2’),

|bs(x) − bt (x)| + ‖σs(x) − σt (x)‖HS ≤ φk(|s − t |), |x | ≤ k.

By employing the cutoff approach, Theorem 1.1 can be extended to include SDEswith local Dini-continuous coefficients, which is presented as below.

Theorem 1.2 Assume (A1’)–(A3’) hold. Then it holds that

limδ→0

E

(sup

0≤t≤T|Xt − Yt |2

)= 0. (1.4)

In particular, if φk(s) = eec0k

4

sα, s ≥ 0, for some α ∈ (0, 1] and c0 > 0, then

E

(sup

0≤t≤T|Xt − Yt |2

)� inf

ε∈(0,1)

{(log log(δ−αε))−

14 + δα(1−ε)

}. (1.5)

Moreover, if σ·(·) is uniformly bounded (i.e., ‖σ‖T ,∞ < ∞), then

E

(sup

0≤t≤T|Xt − Yt |2

)� inf

ε∈(0,1)

{exp

(− 1

CT ‖σ‖2T ,∞(log log(δ−αε))

12

)+ δα(1−ε)

}

(1.6)for some constant CT > 0, where ‖σ‖T ,∞ := sup0≤t≤T ,x∈Rn ‖σt (x)‖HS.

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Under (A1’) and (A2’), (1.1) enjoys a unique strong solution (Xt )t∈[0,T ]; see, forinstance, [22, Theorem 1.1]. Theorem 1.2 has improved the result in [17] since thedrift involved is allowed to be unbounded and local Dini continuous, while the drift in[17] is bounded and Hölder continuous. Furthermore, by comparing (1.5) with (1.6),we infer that the convergence rate of EM scheme is better whenever σ·(·) is uniformlybounded.

Remark 1.3 In fact, in terms of [10, Theorem D], (1.4) holds under (A1’)–(A3’) aswell as the pathwise uniqueness of (1.1), whereas in Sect. 4 we provide an alternativeproof of (1.4) in order to reveal the convergence rate of the EM scheme.

1.3 Degenerate SDEs

So far, most of the existing literature on convergence of EM scheme for SDEs withirregular coefficients is concerned with non-degenerate SDEs; see, e.g., [16–18] forSDEs driven by Brownian motions, and [18] for SDEs driven by jump processes. Theissue for the setup of degenerate SDEs has not yet been considered to date to the bestof our knowledge. Nevertheless, in this subsection, we make an attempt to discuss thetopic for degenerate SDEs with Hölder–Dini continuous drift.

For notation simplicity, we shall write R2n instead of Rn × Rn . Consider the fol-

lowing degenerate SDE on R2n

{dX (1)

t = b(1)t (X (1)

t , X (2)t )dt, X (1)

0 = x (1) ∈ Rn,

dX (2)t = b(2)

t (X (1)t , X (2)

t )dt + σt (X(1)t , X (2)

t )dWt , X (2)0 = x (2) ∈ R

n,(1.7)

where b(1)t , b(2)

t : R2n → R

n , σt : R2n → R

n ⊗ Rn , and (Wt )t≥0 is an n-

dimensional Brownian motion defined on the complete filtered probability space(�,F , (Ft )t≥0,P). (1.7) is also called the stochastic Hamiltonian system, whichhas been investigated extensively in [24,26] on Bismut formulae, in [15] on ergodic-ity, in [21] on hypercontractivity, and in [23] on wellposedness, to name a few. Forapplications of the model (1.7), we refer to, e.g., Soize [20].

Write the gradient operator onR2n as ∇ = (∇(1),∇(2)), where ∇(1) and ∇(2) standfor the gradient operators w.r.t. the first and the second components, respectively.

We assume that there exists φ ∈ Dε ∩ S0 such that for any x = (x (1), x (2)), y =(y(1), y(2)) ∈ R

2n and s, t ∈ [0, T ],(C1) (Hypoellipticity) (∇(2)b(1)

t )(x), σt (x) ∈ Mnnon, and

‖b(1)‖T ,∞ + ‖b(2)‖T ,∞ + ‖∇(2)b(1)‖T ,∞ +∥∥∥(∇(2)b(1))−1

∥∥∥T ,∞

+ ‖σ‖T ,∞ + ‖∇σ‖T ,∞ + ‖σ−1‖T ,∞ < ∞;

(C2) (Regularity of b(1) w.r.t. spatial variables)

|b(1)t (x) − b(1)

t (y)| ≤ |x (1) − y(1)| 23 φ(|x (1) − y(1)|) if x (2) = y(2),

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‖(∇(2)b(1)t )(x) − (∇(2)b(1)

t )(y)‖HS ≤ φ(|x (2) − y(2)|) if x (1) = y(1);

(C3) (Regularity of b(2) w.r.t. spatial variables)

|b(2)t (x) − b(2)

t (y)| ≤ |x (1) − y(1)| 23 φ(|x (1) − y(1)|) + φ72 (|x (2) − y(2)|);

(C4) (Regularity of b(1), b(2) and σ w.r.t. time variables)

|b(1)t (x) − b(1)

s (x)| + |b(2)t (x) − b(2)

s (x)| + ‖σt (x) − σs(x)‖HS ≤ φ(|t − s|).

Observe from (C2) and (C3) that b(1)(·, x (2)) and b(2)(·, x (2)) with fixed x (2) arelocally Hölder–Dini continuous of order 2

3 , and (∇(2)b(1))(x (1), ·) and b(2)(x (1), ·)with fixed x (1) are merely Dini-continuous.

The continuous-time EM scheme associated with (1.7) is as follows:

{dY (1)

t = b(1)tδ (Y (1)

tδ ,Y (2)tδ )dt, X (1)

0 = x (1) ∈ Rn,

dY (2)t = b(2)

tδ (Y (1)tδ ,Y (2)

tδ )dt + σtδ (Y(1)tδ ,Y (2)

tδ )dWt , X (2)0 = x (2) ∈ R

n .(1.8)

Another contribution in this paper reads as below.

Theorem 1.4 Let (C1)–(C4) hold. Then

(E

(sup

0≤t≤T|Xt − Yt |2

))1/2

�T φ(CT√

δ)

for some constant CT ≥ 1, in which

Xt :=(X (1)t

X (2)t

)and Yt :=

(Y (1)t

Y (2)t

).

According to [23, Theorem 1.2], (1.7) admits a unique strong solution under theassumptions (C1)–(C3). In fact, (1.7) is wellposed under (C1)–(C3)withφ ∈ D0∩S0in lieu of φ ∈ Dε ∩ S0. Nevertheless, the requirement φ ∈ Dε ∩ S0 is imposed inorder to reveal the order of convergence for the EM scheme above. By applying thecutoff approach and refining the argument of [23, Theorem 2.3] (see also Lemma 5.1below), the boundedness of coefficients can be removed. We herein do not go intodetails since the corresponding trick is quite similar to the proof of Theorem 1.2.

The outline of this paper is organized as follows: In Sect. 2, we elaborate regularityof non-degenerate Kolmogorov equation, which plays an important role in dealingwith convergence rate of EM scheme for non-degenerate SDEs with Hölder–Dinicontinuous and unbounded drifts; In Sects. 3, 4 and 5, we complete the proofs ofTheorems 1.1, 1.2 and 1.4, respectively.

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2 Regularity of Non-degenerate Kolmogorov Equation

Let (ei )i≥1 be an orthogonal basis of Rn . For any λ > 0, consider the followingRn-valued parabolic equation:

∂t uλt + Ltu

λt + bt + ∇bt u

λt = λuλ

t , uλT = 0n, (2.1)

where ∇bt uλt means the directional derivative along the direction bt , 0n is the zero

vector in Rn and

Lt := 1

2

∑i, j

⟨(σtσ

∗t )(·)ei , e j

⟩∇ei ∇e j

with σ ∗t standing for the transpose of σt . Let (P0

s,t )0≤s≤t be the semigroup generatedby (Zs,x

t )0≤s≤t which solves an SDE below

dZs,xt = σt (Z

s,xt )dWt , t > s, Zs,x

s = x . (2.2)

By the chain rule, it follows from (2.1) that

∂t

(e−λ(t−s)P0

s,t uλt

)= e−λ(t−s)

{−λP0

s,t uλt + P0

s,t Ltuλt + P0

s,t∂t uλt

}

= −e−λ(t−s)P0s,t

{bt + ∇bt u

λt

}.

Thus, integrating from s to T and taking advantage of uλT = 0n, we arrive at

uλs =

∫ T

se−λ(t−s)P0

s,t

{bt + ∇bt u

λt

}dt . (2.3)

For notation simplicity, let

�T ,σ = eT2 ‖∇σ‖2T ,∞‖σ−1‖T ,∞ (2.4)

and

�T ,σ = 48e288 T2‖∇σ‖4T ,∞

{6√2eT ‖∇σ‖2T ,∞‖σ−1‖4T ,∞ + T ‖∇σ−1‖2T ,∞

+ 2T 2‖∇2σ‖2T ,∞‖σ−1‖2T ,∞e2T ‖∇σ‖2T ,∞}

.(2.5)

Moreover, set

ϒT ,σ :=√

�T ,σ

{3 + 2‖b‖T ,∞ + 28

(�T ,σ +

√�T ,σ

)‖b‖2T ,∞

}. (2.6)

The lemma below plays a crucial role in investigating error analysis.

Lemma 2.1 Under (A1) and (A2), for anyλ ≥ 9π�2T ,σ ‖b‖2T ,∞+4(‖b‖T ,∞+�T ,σ )2,

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(i) (2.1) (i.e., (2.3)) enjoys a unique strong solution uλ ∈ C([0, T ];C1b(R

n;Rn));(ii) ‖∇uλ‖T ,∞ ≤ 1

2 ;

(iii) ‖∇2uλ‖T ,∞ ≤ ϒT ,σ

∫ T0

e−λt

t φ(‖σ‖T ,∞√t)dt, where φ(s) := √

φ2(s) + s,s ≥ 0.

Proof To show (i)–(iii), it boils down to refine the argument of [22, Lemma 2.1]. (i)holds for any λ ≥ 4(‖b‖T ,∞ + �T ,σ )2 via the Banach fixed-point theorem.

In what follows, we aim to show (ii) and (iii) hold true, one-by-one. Observe from[12, Theorem 3.1, p.218] that

d∇ηZs,xt =

(∇∇ηZ

s,xt

σt

) (Zs,xt)dWt , t ≥ s, ∇ηZ

s,xs = η ∈ R

n . (2.7)

Using Itô’s isometry and Gronwall’s inequality, one has

E|∇ηZs,xt |2 ≤ |η|2eT ‖∇σ‖2T ,∞ . (2.8)

Utilizing the BDG inequality, we deduce that

E|∇ηZs,xt |4 ≤ 8

{|η|4 + 36(t − s)‖∇σ‖4T ,∞

∫ t

sE|∇ηZ

s,xu |4du

},

which, combining with Gronwall’s inequality, yields that

E|∇ηZs,xt |4 ≤ 8|η|4e288 T 2‖∇σ‖4T ,∞ . (2.9)

Recall from [22, (2.8)] that the following Bismut formula

∇ηP0s,t f (x) = E

(f(Zs,xt)

t − s

∫ t

s

⟨σ−1r

(Zs,xr

)∇ηZs,xr , dWr

⟩), f ∈ Bb(R

n)

(2.10)holds. By the Cauchy–Schwartz inequality, the Itô isometry and (2.8), we obtain that

|∇ηP0s,t f |2(x) ≤ �2

T ,σ |η|2P0s,t f

2(x)

t − s, f ∈ Bb(R

n), (2.11)

where �T ,σ > 0 is defined in (2.4). So, one infers from (2.3) and (2.11) that

‖∇uλs ‖ ≤

∫ T

se−λ(t−s)‖∇P0

s,t {bt + ∇bt uλt }‖dt

≤ �T ,σ

(1 + ‖∇uλ‖T ,∞

) ‖b‖T ,∞∫ T

0

e−λt

√tdt

≤ λ− 12√

π�T ,σ ‖b‖T ,∞(1 + ‖∇uλ‖T ,∞).

Thus, (ii) follows by taking λ ≥ 9π�2T ,σ ‖b‖2T ,∞.

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In the sequel, we intend to verify (iii). Set γs,t := ∇η∇η′ Zs,xt for any η, η′ ∈ R

n .Notice from (2.7) that

dγs,t ={(∇γs,tσt

) (Zs,xt)+

(∇∇ηZ

s,xt

∇∇η′ Zs,xt

σt

) (Zs,xt)}

dWt , t ≥ s, γs,s = 0n.

By the Doob submartingale inequality and the Itô isometry, besides the Gronwallinequality and (2.8), we derive that

sups≤t≤T

E|γs,t |2 ≤ 16T ‖∇2σ‖2T ,∞e288T2‖∇σ‖4T ,∞+2T ‖∇σ‖2T ,∞|η|2|η′|2. (2.12)

From (2.10) and the Markov property, we have

∇ηP0s,t f (x) = E

⎛⎜⎝(P0

t+s2 ,t

f) (

Zs,xt+s2

)(t − s)/2

∫ t+s2

s

⟨σ−1r

(Zs,xr

)∇ηZs,xr , dWr

⟩⎞⎟⎠ .

This further gives that

1

2

(∇η′∇ηP

0s,t f

)(x)

= E

⎛⎜⎜⎝

(∇∇η′ Zs,x

t+s2

P0t+s2 ,t

f

)(Zs,x

t+s2

)

t − s

∫ t+s2

s

⟨σ−1r

(Zs,xr

)∇ηZs,xr , dWr

⟩⎞⎟⎟⎠

+ E

⎛⎜⎝(P0

t+s2 ,t

f) (

Zs,xt+s2

)t − s

∫ t+s2

s

⟨(∇∇η′ Zs,x

rσ−1r

) (Zs,xr

)∇ηZs,xr , dWr

⟩⎞⎟⎠

+ E

⎛⎜⎝(P0

t+s2 ,t

f) (

Zs,xt+s2

)t − s

∫ t+s2

s

⟨σ−1r

(Zs,xr

)∇η′∇ηZs,xr , dWr

⟩⎞⎟⎠ .

Thus, applying Cauchy–Schwartz’s inequality and Itô’s isometry and taking (2.9),(2.11) and (2.12) into consideration, we derive that

|∇η′∇ηP0s,t f |2(x)

≤ 12

⎧⎪⎨⎪⎩6‖σ

−1‖2T ,∞E

∣∣∣∇P0t+s2 ,t

f∣∣∣2 (Zs,x

t+s2

)(t − s)5/2

×(E

∣∣∣∇η′ Zs,xt+s2

∣∣∣4)1/2

(∫ t+s2

sE∣∣∇ηZ

s,xr

∣∣4 dr)1/2

123


+ P0s,t f

2(x)

(t − s)2‖∇σ−1‖2T ,∞

∫ t+s2

s

(E|∇η′ Zs,x

r |4)1/2 (

E|∇ηZs,xr |4

)1/2dr

+ P0s,t f

2(x)

(t − s)2‖σ−1‖2T ,∞

∫ t+s2

sE∣∣∇η′∇ηZ

s,xr

∣∣2 dr}

≤ �T ,σ |η|2|η′|2 P0s,t f

2(x)

(t − s)2, (2.13)

where �T ,σ > 0 is defined as in (2.5).Set f (·) := f (·) − f (x) for fixed x ∈ R

n and f ∈ Bb(Rn) which verifies

| f (x) − f (y)| ≤ φ(|x − y|), x, y ∈ Rn (2.14)

for some φ ∈ D . For f ∈ Bb(Rn) such that (2.14), (2.13) implies that

|∇η′∇ηP0s,t f |2(x) = |∇η′∇ηP

0s,t f |2(x) ≤ �T ,σ |η|2|η′|2

(t − s)2E| f (Zs,x

t ) − f (x)|2

≤ �T ,σ |η|2|η′|2(t − s)2

φ2(‖σ‖T ,∞(t − s)1/2),

(2.15)where in the second display we have used that

Zs,xt − x =

∫ t

sσr (Z

s,xr )dWr ,

and utilized Jensen’s inequality as well as Itô’s isometry.Let ft = bt + ∇bt u

λt . For any λ ≥ 9π�2

T ,σ ‖b‖2T ,∞ + 4(‖b‖T ,∞ + �T ,σ )2, notefrom (ii), (2.11) and (2.13) that

| ft (x) − ft (y)| ≤ (1 + ‖∇uλ‖T ,∞)φ(|x − y|)+‖b‖T ,∞‖∇uλ

t (x) − ∇ut (y)‖1{|x−y|≥1}+‖b‖T ,∞‖∇uλ

t (x) − ∇ut (y)‖1{|x−y|≤1}

≤ 3

2φ(|x − y|) + ‖b‖T ,∞

√|x − y|1{|x−y|≥1}

+10

(�T ,σ +

√�T ,σ

)‖b‖2T ,∞

√|x − y|√|x − y|

× log

(e + 1

|x − y|)1{|x−y|≤1}

≤{3 + 2‖b‖T ,∞ + 28

(�T ,σ +

√�T ,σ

)‖b‖2T ,∞

}φ(|x − y|)

with φ(s) := √φ2(s) + s, s ≥ 0, where in the second inequality we have used

[22, Lemma 2.2 (1)], and the fact that the function [0, 1] � x �→ √x log(e + 1

x ) isnon-decreasing. As a result, (iii) follows from (2.15). ��

123


3 Proof of Theorem 1.1

With Lemma 2.1 in hand, we now in the position to complete the

Proof of Theorem 1.1 Throughout the whole proof, we assume λ ≥ 9π�2T ,σ ‖b‖2T ,∞ +

4(‖b‖T ,∞ + �T ,σ )2 so that (i)–(iii) in Lemma 2.1 hold. For any t ∈ [0, T ], applyingItô’s formula to x + uλ

t (x), x ∈ Rn , we deduce from (2.1) that

Xt + uλt (Xt )= x + uλ

0(x)+λ

∫ t

0uλs (Xs)ds+

∫ t

0{In×n + (∇uλ

s )(·)}(Xs)σs(Xs)dWs,

(3.1)where In×n is an n × n identity matrix, and that

Yt + uλt (Yt ) = x + uλ

0(x)+λ

∫ t

0uλs (Ys)ds+

∫ t

0{In×n + (∇uλ

s )(·)}(Ys)σsδ (Ysδ )dWs

+∫ t

0{In×n + (∇uλ

s )(·)}(Ys){bsδ (Ysδ ) − bs(Ys)}ds

+ 1

2

∫ t

0

∑k, j

⟨{(σsδ σ ∗sδ )(Ysδ ) − (σsσ

∗s )(Ys)}ek, e j

⟩(∇ek∇e j u

λs )(Ys)ds.

(3.2)For notation simplicity, set

Mλt := Xt − Yt + uλ

t (Xt ) − uλt (Yt ). (3.3)

Using the elementary inequality: (a+b)2 ≤ (1+ε)(a2+ε−1b2) for arbitrary ε, a, b >

0, we derive from (ii) that

|Xt − Yt |2 ≤ (1 + ε)(|Mλt |2 + ε−1|uλ

t (Xt ) − uλt (Yt )|2)

≤ (1 + ε)

(|Mλ

t |2 + ε−1

4|Xt − Yt |2

).

In particular, taking ε = 1 leads to

|Xt − Yt |2 ≤ 1

2|Xt − Yt |2 + 2|Mλ

t |2.

As a consequence,

E

(sup

0≤s≤t|Xs − Ys |2

)≤ 4E

(sup

0≤s≤t|Mλ

s |2)

. (3.4)

Inwhat follows, our goal is to estimate the termon the right-hand side of (3.4). Observefrom the definition of the Hilbert–Schmidt norm that

123


∫ t

0E

∣∣∣∣∣∣∑k, j

〈{(σsδ σ ∗sδ )(Ysδ ) − (σsσ

∗s )(Ys)}ek, e j 〉(∇ek∇e j u

λs )(Ys)

∣∣∣∣∣∣2

ds

�T ‖∇2uλ‖2T ,∞∫ t

0E‖(σsδ σ ∗

sδ )(Ysδ ) − (σsσ∗s )(Ys)‖2HSds. (3.5)

Thus, by Hölder’s inequality, Doob’s submartingale inequality and Itô’s isometry, itfollows from (3.1), (3.2) and (3.5) that

E

(sup

0≤s≤t|Mλ

s |2)

≤ CT

{λ2∫ t

0E|uλ

s (Xs) − uλs (Ys)|2ds

+ (1 + ‖∇u‖2T ,∞)

∫ t

0E|bsδ (Ys) − bsδ (Ysδ )|2ds

+ (1 + ‖∇u‖2T ,∞)

∫ t

0E|bs(Ys) − bsδ (Ys)|2ds

+∫ t

0E‖{(∇uλ

s )(Xs) − (∇uλs )(Ys)}σs(Xs)‖2HSds

+ (1 + ‖∇u‖2T ,∞)

∫ t

0E‖σsδ (Xs) − σsδ (Ysδ )‖2HSds

+ ‖∇2uλ‖2T ,∞∫ t

0E‖{σsδ (Ys) − σsδ (Ysδ )}σ ∗

sδ (Ysδ )‖2HSds

+ ‖∇2uλ‖2T ,∞∫ t

0E‖σs(Ys){σ ∗

sδ (Ys) − σ ∗sδ (Ysδ )}‖2HSds

+ (1 + ‖∇u‖2T ,∞)

∫ t

0E‖σs(Xs) − σsδ (Xs)‖2HSds

+ ‖∇2uλ‖2T ,∞∫ t

0E‖σs(Ys){σ ∗

s (Ys) − σ ∗sδ (Ys)}‖2HSds

+‖∇2uλ‖2T ,∞∫ t

0E‖{σs(Ys) − σsδ (Ys)}σ ∗

sδ (Ysδ )‖2HSds}

=: CT

(10∑i=1

Ii (t)

)

for some constant CT > 0. Also, applying Hölder’s inequality and Itô’s isometry, wededuce from (A1) that

E|Yt − Ytδ |2 ≤ βT δ (3.6)

for some constant βT ≥ 1. By Taylor’s expansion, it is obvious to see that

I1(t) + I4(t) � {λ2‖∇uλ‖2T ,∞ + ‖∇2uλ‖2T ,∞‖σ‖2T ,∞}∫ t

0E|Xs − Ys |2ds. (3.7)

123


From (A3) and due to the fact that φ(·) is increasing and δ ∈ (0, 1), one has

I3(t) +10∑i=8

Ii (t) �T {1 + ‖∇uλ‖2T ,∞ + ‖∇2uλ‖2T ,∞‖σ‖2T ,∞}φ2(√

δ). (3.8)

In view of (A2), we derive that

I2(t) +7∑

i=5

Ii (t)

� {1 + ‖∇uλ‖2T ,∞}∫ t

0Eφ(|Ys − Ysδ |)2ds

+ {1 + ‖∇uλ‖2T ,∞}‖∇σ‖2T ,∞∫ t

0E|Xs − Ys |2ds

+ {1 + ‖∇uλ‖2T ,∞ + ‖∇2uλ‖2T ,∞‖σ‖2T ,∞}‖∇σ‖2T ,∞∫ t

0E|Ys − Ysδ |2ds.

(3.9)

Thus, taking (3.6)–(3.9) into account and applying Jensen’s inequality gives that

E

(sup

0≤s≤t|Mλ

s |2)

�T CT ,σ,λ{δ + φ2(βT√

δ)} + CT ,σ,λ

∫ t

0E|Xs − Ys |2ds,

where

CT ,σ,λ := {1 + ‖∇σ‖2T ,∞}{5

4+ (1 + λ2)‖∇2uλ‖2T ,∞‖σ‖2T ,∞

}. (3.10)

Owing to φ ∈ D, we conclude that φ(0) = 0, φ′ > 0 and φ′′ < 0 so that, for anyc > 0 and δ ∈ (0, 1),

φ(cδ) = φ(0) + φ′(ξ)cδ ≥ φ′(c)cδ,

where ξ ∈ (0, cδ). This further implies that

E

(sup

0≤s≤t|Mλ

s |2)

�T CT ,σ,λφ2(βT

√δ) + CT ,σ,λ

∫ t

0E|Xs − Ys |2ds.

Substituting this into (3.4) gives that

E

(sup

0≤s≤t|Xs − Ys |2

)�T CT ,σ,λφ

2(βT√

δ) + CT ,σ,λ

∫ t

0E|Xs − Ys |2ds.

123


Thus, Gronwall’s inequality implies that there exists CT > 0 such that

E

(sup

0≤s≤t|Xs − Ys |2

)≤ CT CT ,σ,λe

CT CT ,σ,λφ2(βT√

δ). (3.11)

So the desired assertion holds immediately. ��


We shall adopt the cutoff approach to finish the

Proof of Theorem 1.2 Take ψ ∈ C∞b (R+) such that 0 ≤ ψ ≤ 1, ψ(r) = 1 for r ∈

[0, 1] and ψ(r) = 0 for r ≥ 2. For any t ∈ [0, T ] and k ≥ 1, define the cutofffunctions

b(k)t (x) = bt (x)ψ(|x |/k) and σ

(k)t (x) = σt (ψ(|x |/k)x), x ∈ R

n .

It is easy to see that b(k) and σ (k) satisfy (A1). For fixed k ≥ 1, consider the followingSDE

dX (k)t = b(k)

t (X (k)t )dt + σ

(k)t (X (k)

t )dWt , t > 0, X (k)0 = X0 = x . (4.1)

The corresponding continuous-time EM of (4.1) is defined by

dY (k)t = b(k)

tδ (Y (k)tδ )dt + σ

(k)tδ (Y (k)

tδ )dWt , t > 0, Y (k)0 = X0 = x . (4.2)

Applying BDG’s inequality, Hölder’s inequality andGronwall’s inequality, we deducefrom (A1’) that

E

(sup

0≤t≤T|Xt |4

)+E

(sup

0≤t≤T|Yt |4

)+E

(sup

0≤t≤T|X (k)

t |4)

+E

(sup

0≤t≤T|Y (k)

t |4)

≤ CT

(4.3)for some constant CT > 0. Note that

E

(sup

0≤t≤T|Xt − Yt |2

)≤ 3E

(sup

0≤t≤T|Xt − X (k)

t |2)

+ 3E

(sup

0≤t≤T|X (k)

t − Y (k)t |2

)

+ 3E

(sup

0≤t≤T|Yt − Y (k)

t |2)

=: I1 + I2 + I3.

123


For the terms I1 and I3, in terms of the Chebyshev inequality we find from (4.3) that

I1 + I3 � E

(sup

0≤t≤T|Xt − X (k)

t |21{sup0≤t≤T |Xt |≥k}

)

+ E

(sup

0≤t≤T|Yt − Y (k)

t |21{sup0≤t≤T |Yt |≥k}

)

�

√√√√E

(sup

0≤t≤T|Xt |4

)+ E

(sup

0≤t≤T|X (k)

t |4)√

E(sup0≤t≤T |Xt |2

)k

+√√√√E

(sup

0≤t≤T|Yt |4

)+ E

(sup

0≤t≤T|Y (k)

t |4)√

E(sup0≤t≤T |Yt |2

)k

�T1

k,

(4.4)

where in the first display we have used the facts that {Xt �= X (k)t } ⊂ {sup0≤s≤t |Xs | ≥

k} and {Yt �= Y (k)t } ⊂ {sup0≤s≤t |Ys | ≥ k}. Observe from (A1’) that 9π�2

T ,σ (k)

‖b(k)‖2T ,∞ + 4(‖b(k)‖T ,∞ + �T ,σ (k) )2 ≤ ec k2for some c > 0. Next, according to

(3.11), by taking λ = ec k2there exits CT > 0 such that

I2 ≤ eCTCT ,σ (k),λφ2k (βT

√δ). (4.5)

Herein, CT ,σ (k),λ > 0 is defined as in (3.10) with σ and uλ replaced by σ (k) and uλ,k ,respectively, where uλ,k solves (2.3) by writing b(k) instead of b. Consequently, weconclude that

E

(sup

0≤t≤T|Xt − Yt |2

)≤ c0

k+ c0e

CTCT ,σ (k),λφ2k (βT

√δ) (4.6)

for some c0 > 0. For any ε > 0, taking k = �2c0/ε and letting δ go to zero impliesthat

limδ→0

E

(sup

0≤t≤T|Xt − Yt |2

)≤ ε.

Thus, (1.4) follows due to the arbitrariness of ε.

For φk(s) = eec0k

4

sα, s ≥ 0, with α ∈ (0, 1], we deduce from Lemma 2.1 (iii) that

‖∇2uλ,k‖T ,∞ ≤ 1

2(4.7)

123


whenever

λ ≥{2ϒT ,σ (k)

(ee

c0k4 ‖σ (k)‖α

T ,∞�(α/2) + ‖σ (k)‖1/2T ,∞�(1/4)

)}2/α

+ 9π(�T ,σ (k) )2‖b(k)‖2T ,∞ + 4(‖b(k)‖T ,∞ + �T ,σ (k) )

2.

(4.8)

Since the right-hand side of (4.8) can be bounded by eeCT k4

for some constant CT > 0

due to (A1’), we can take λ = eeCT k4

so that (4.7) holds. Thus, (4.6), together with(4.7) and (A1’), yields that

E

(sup

0≤t≤T|Xt − Yt |2

)≤ CT

k+ CT e

eCT k4

δα

for some constants CT , CT > 0. Thus, (1.5) follows immediately by taking

k =⌊(

1

CTlog log δ−αε

) 14⌋

. (4.9)

Next, we aim to show that (1.6) holds true. In view of (4.3) and (4.4), it followsfrom Hölder’s inequality that

I1 + I3 �

√√√√E

(sup

0≤t≤T|Xt − X (k)

t |4)√√√√

P

(sup

0≤t≤T|Xt | ≥ k

)

+√√√√E

(sup

0≤t≤T|Yt − Y (k)

t |4)√√√√

P

(sup

0≤t≤T|Yt | ≥ k

)

�T

√√√√P

(sup

0≤t≤T|Xt | ≥ k

)+√√√√P

(sup

0≤t≤T|Yt | ≥ k

).

(4.10)

By (A1’), we infer that

sup0≤s≤t

|Ys | ≤ |x | + KT T + sup0≤s≤t

|Nt | + KT

∫ t

0|Ysδ |ds

≤ |x | + KT T + sup0≤s≤t

|Nt | + KT

∫ t

0sup

0≤r≤s|Yr |ds

(4.11)

where

Nt :=∫ t

0σsδ (Ysδ )dWs .

123


Thus, Gronwall’s inequality enables us to get that

sup0≤s≤t

|Ys | ≤ (|x | + KT T )eKT T + eKT T sup0≤s≤t

|Ns |.

For any integer k ≥ 1 such that

ρ := ke−KT T − |x | − KT T > 0,

we derive from (4.11) that

P

(sup

0≤t≤T|Yt | ≥ k

)= P

(sup

0≤t≤T|Nt | ≥ ρ

).

This, by taking advantage of [19, Proposition 6.8], yields that

P

(sup

0≤t≤T|Yt | ≥ k

)= P

(〈N 〉T ≤ ‖σ‖2∞T , sup

0≤t≤T|Nt | ≥ ρ

)

≤ 2n exp

(− ρ2

2n‖σ‖2T ,∞T

),

(4.12)

where 〈N 〉t stands for the quadratic variation process of Nt . Next, by using the inequal-ity: (a − b)2 ≥ 1

2a2 − b2, a, b ∈ R, we deduce from (4.12) that

P

(sup

0≤t≤T|Yt | ≥ k

)≤ 2n exp

((|x | + KT T )2

2n‖σ‖2T ,∞T

)exp

(− k2

4n‖σ‖2T ,∞T e2KT T

).

(4.13)Similarly, one can obtain that

P

(sup

0≤t≤T|Xt | ≥ k

)≤ 2n exp

((|x | + KT T )2

2n‖σ‖2T ,∞T

)exp

(− k2


).

(4.14)Inserting (4.13) and (4.14) back into (4.10) leads to

I1 + I3 �T exp

(− k2


).

This, together with (4.5), (4.7) and (A1’), gives that

E

(sup

0≤t≤T|Xt − Yt |2

)≤ CT exp

(− k2


)+ CT e

eCT k4

δα

123


for some constants CT , CT > 0. As a consequence, (1.6) follows by taking k given in(4.9). ��


For simplicity, for any f : Rm1 → Rm2 , let

[ f ]L = supx �=y

| f (x) − f (y)||x − y| , ‖ f ‖∞ = sup

x∈Rm1

| f (x)|.

The proof of Theorem 1.4 relies on regularization properties of the following R2n-

valued degenerate parabolic equation

∂t uλt + L b,σ

t uλt + bt = λuλ

t , uλT = 02n, t ∈ [0, T ], λ > 0, (5.1)

where 02n is the zero vector in R2n ,

bt :=(b(1)t

b(2)t

)and L b,σ

t uλ := 1

2

n∑i, j=1

〈(σtσ ∗t )(·)ei , e j 〉∇(2)

ei ∇(2)e j u

λ +∇(1)

b(1)tuλ +∇(2)

b(2)tuλ.

The following lemma on regularity estimate of solution to (5.1) is taken from [23,Theorem 3.10, (4.4)] and is an essential ingredient in analyzing numerical approxi-mation.

Lemma 5.1 Under (C1)–(C3), (5.1) has a unique solution uλ ∈ C([0, T ];C1b

(R2n;R2n)) such that for all t ∈ [0, T ],

‖∇uλt ‖∞ + ‖∇(2)∇(2)uλ

t ‖∞ + [∇(2)ut ]L ≤ C∫ T

0e−λt φ(t

12 )

tdt, (5.2)

where C > 0 is a constant.

From now on, we move forward to complete the

Proof of Theorem 1.4 For notation simplicity, set

Xt :=(X (1)t

X (2)t

), Yt :=

(Y (1)t

Y (2)t

)and bt (x) :=

(b(1)t (x)

b(2)t (x)

), x ∈ R

2n .

Then (1.7) and (1.8) can be reformulated, respectively, as

dXt = bt (Xt )dt +(0n×nσt

)(Xt )dWt , t > 0, X0 = x =

(x1x2

)∈ R

2n,

123


where 0n×n is an n × n zero matrix, and

dYt = btδ (Ytδ )dt +(0n×nσtδ

)(Ytδ )dWt , t > 0, Y0 = x ∈ R

2n .

Note from (5.2) that there exists λ0 > 0 sufficiently large such that for any t ∈ [0, T ],

‖∇uλt ‖∞ + ‖∇(2)∇(2)uλ

t ‖∞ + [∇(2)uλt ]L ≤ 1

2, λ ≥ λ0. (5.3)

Applying Itô’s formula to x + uλt (x) for any x ∈ R

2n , we deduce that

Xt + uλt (Xt ) = x + uλ

0(x) + λ

∫ t

0uλs (Xs)ds +

∫ t

0

(0n×nσs

)(Xs)dWs

+∫ t

0

(∇(2)

σsdWsuλs

)(Xs), (5.4)

and that

Yt + uλt (Yt ) = x + uλ

0(x) + λ

∫ t

0uλs (Ys)ds

+∫ t

0{I2n×2n + (∇us)(·)}(Ys){bsδ (Ysδ ) − bs(Ys)}ds

+∫ t

0

(0n×nσsδ

)(Ysδ )dWs +

∫ t

0

(∇(2)

σsδ (Ysδ )dWsuλs

)(Ys)

+ 1

2

∫ t

0

n∑k, j=1

〈{(σsδ σ ∗sδ )(Ysδ

)− (σsσ ∗s

)(Ys)

}ek , e j 〉

(∇(2)ek ∇(2)

e j uλs

)(Ys)ds,

(5.5)where I2n×2n is an 2n × 2n identity matrix. Thus, using Hölder’s inequality, Doob’ssubmartingale inequality and Itô’s isometry and taking (3.5) into consideration givesthat

E

(sup

0≤s≤t|Mλ

s |2)

≤ C0,T

{∫ t

0E|uλ

s (Xs) − uλs (Ys)|2ds

+(1 + ‖∇uλ‖2T ,∞

) ∫ t

0E|bsδ (Ys) − bsδ (Ysδ )|2ds

+(1 + ‖∇uλ‖2T ,∞

) ∫ t

0E|bs(Ys) − bsδ (Ys)|2ds

+∫ t

0E‖{(∇(2)uλ

s )(Xs) − ∇(2)uλs (Ys)}σs(Xs)‖2HSds

+(1 + ‖∇(2)uλ‖2T ,∞

) ∫ t

0E‖σsδ (Xs) − σsδ (Ysδ )‖2HSds

+ (1 + ‖∇(2)uλ‖2T ,∞)

∫ t

0E‖σs(Xs) − σsδ (Xs)‖2HSds

123


+ ‖∇(2)∇(2)uλ‖2T ,∞∫ t

0E‖{σsδ (Ys) − σsδ (Ysδ )}σ ∗

sδ (Ysδ )‖2HSds

+ ‖∇(2)∇(2)uλ‖2T ,∞∫ t

0E‖σs(Ys){σ ∗

sδ (Ys) − σ ∗sδ (Ysδ )}‖2HSds

+ ‖∇(2)∇(2)uλ‖2T ,∞∫ t

0E‖σs(Ys){σ ∗

s (Ys) − σ ∗sδ (Ys)}‖2HSds

+‖∇(2)∇(2)uλ‖2T ,∞∫ t

0E‖{σs(Ys) − σsδ (Ys)}σ ∗

sδ (Ysδ )‖2HSds}

=: C0,T

(10∑i=1

Ji (t)

)

for some constant C0,T > 0, where Mλt is defined as in (3.3). By using Hölder’s

inequality and the BDG inequality, (C1) implies that

E|Yt − Ytδ |p � δp2 , p ≥ 1. (5.6)

Utilizing Taylor’s expansion, one gets from (3.6), (5.3) and (5.6) that

J1(t) + J4(t) + J5(t) �{1 + ‖∇uλ‖2T ,∞ + ‖∇∇(2)uλ‖2T ,∞‖σ‖2T ,∞

} ∫ t

0E|Xs − Ys |2ds

+ {1 + ‖∇(2)uλ‖2T ,∞}∫ t

0E|Ys − Ysδ |2ds

� δ +∫ t

0E|Xs − Ys |2ds.

Next, (C1), (C5) and (5.3) yield that

J3(t) + J6(t) + J9(t) + J10(t) � φ2(√

δ),

where we have also used that φ(·) is increasing and δ ∈ (0, 1). Additionally, by virtueof (C1), (C2), and (5.3), we infer from (C3) that

J2(t) + J7(t) + J8(t) � δ +∫ t

0E|bsδ

(Y (1)s ,Y (2)

s

)− bsδ

(Y (1)sδ ,Y (2)

s

)|2ds

+∫ t

0E

∣∣∣bsδ(Y (1)sδ ,Y (2)

s

)− bsδ

(Y (1)sδ ,Y (2)

sδ

)∣∣∣2 ds≤ C1,T

{δ +

∫ t

0E

∣∣∣b(1)sδ (Y (1)

s ,Y (2)s ) − b(1)

sδ (Y (1)sδ ,Y (2)

s )

∣∣∣2 ds+∫ t

0E

∣∣∣b(2)sδ (Y (1)

s ,Y (2)s ) − b(2)

sδ (Y (1)sδ ,Y (2)

s )

∣∣∣2 ds+∫ t

0E

∣∣∣b(1)sδ (Y (1)

sδ ,Y (2)s ) − b(1)

sδ (Y (1)sδ ,Y (2)

sδ )

∣∣∣2 ds

123


+∫ t

0E

∣∣∣b(2)sδ (Y (1)

sδ ,Y (2)s ) − b(2)

sδ (Y (1)sδ ,Y (2)

sδ )

∣∣∣2 ds}

=: C1,T

(δ +

4∑i=1

�i (t)

)

for some constant C1,T > 0. From (C2), (C3), (5.6) and φ ∈ Dε, we derive fromHölder’s inequality and Jensen’s inequality that

�1(t) + �2(t) �2∑

i=1

∫ t

0E

⎛⎜⎜⎝∣∣∣b(i)

sδ (Y (1)s ,Y (2)

s ) − b(i)sδ (Y (1)

sδ ,Y (2)s )

∣∣∣∣∣∣Y (1)

s − Y (1)sδ

∣∣∣23φ(∣∣∣Y (1)

s − Y (1)sδ

∣∣∣)1{Y(1)

s �=Y(1)s }

×∣∣∣Y (1)

s − Y (1)sδ

∣∣∣23φ(∣∣∣Y (1)

s − Y (1)sδ

∣∣∣)⎞⎟⎟⎠

2

ds

�∫ t

0E

(∣∣∣Y (1)s − Y (1)

sδ

∣∣∣23φ(∣∣∣Y (1)

s − Y (1)sδ

∣∣∣))2

ds

�∫ t

0

(Eφ(∣∣∣Y (1)

s − Y (1)sδ

∣∣∣)2(1+ε)) 1

1+ε

(E

∣∣∣Y (1)s −Y (1)

sδ

∣∣∣4(1+ε)

3ε

) ε1+ε

ds

� δ23 φ2(C2,T

√δ)

(5.7)for some constant C2,T > 0. With regard to the term �3(t), (C1) and (5.6) lead to

�3(t) � ‖∇(2)b(1)‖2T ,∞∫ t

0E

∣∣∣Y (1)s − Y (1)

sδ

∣∣∣2 ds � δ. (5.8)

Due to (C3), observe from Jensen’s inequality and (5.6) that

�4(t) �∫ t

0E

(|b(2)

sδ (Y (1)sδ , Y (2)

s ) − b(2)sδ (Y (1)

sδ , Y (2)sδ )|

φ(|Y (2)s − Y (2)

sδ |)1{Y (2)

s �=Y (2)sδ } × φ(|Y (2)

s − Y (2)sδ |)

)2

ds

�∫ t

0Eφ(|Y (2)

s − Y (2)sδ |)2ds

� φ2(C3,T√

δ)

for some constant C3,T > 0. Consequently, we arrive at

E

(sup

0≤s≤t|Xs − Ys |2

)�T φ2(C4,T

√δ) +

∫ t

0E sup

0≤r≤s|Xr − Yr |2ds

for some constant C4,T ≥ 1. Thus, the desired assertion follows from the Gronwallinequality. ��

123


Acknowledgements We are indebted to the referee, the associate editor and Professor Feng-Yu Wang forvaluable comments and suggestions which have greatly improved our paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

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convergence rate of euler–maruyama scheme for sdes with ...term b satisﬁes the global...

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